RECURSIVE VERSIONS OF THE LEVENBERG-MARQUARDT REASSIGNED SPECTROGRAM AND OF THE SYNCHROSQUEEZED STFT

Transcription

1 RECURSIVE VERSIONS OF THE LEVENBERG-MARQUARDT REASSIGNED SPECTROGRAM AND OF THE SYNCHROSQUEEZED STFT Dominique Fourer 1, Françoi Auger 1, Patrick Flandrin 2 1 LUNAM Univerity, IREENA, Saint-Nazaire, France 2 Laboratoire de Phyique de l École Normale Supérieure de Lyon, CNRS and Univerité de Lyon, France dominique.fourer@univ-nante.fr, francoi.auger@univ-nante.fr, patrick.flandrin@en-lyon.fr ABSTRACT In thi paper, we firt preent a recurive implementation of a recently propoed reaignment proce called the Levenberg- Marquardt reaignment, which allow a uer to adjut the limne of the ignal component localization in the timefrequency plane. Thank to a generalization of the ignal recontruction formula, we alo preent a recurive implementation of the ynchroqueezed hort-time Fourier tranform. Thi approach pave the way for a real-time computation of a reverible and adjutable almot-ideal time-frequency repreentation. Inde Term time-frequency analyi, reaignment, ynchroqueezing, hort-time Fourier tranform, recurive filtering. 1. INTRODUCTION Time-frequency analyi aim at epanding a ignal into a et of non-tationary component and can be performed uing the well-known Short-Time Fourier Tranform STFT. However, thi tool uffer from a poor energy localization in the time-frequency plane. A poible improvement wa propoed by the reaignment method, introduced by Kodera et al.[1] and generalized by Auger and Flandrin to any time-frequency ditribution of the Cohen and affine clae in [2]. Recently, Auger et al.[3] propoed an etenion of thi method inpired by the Levenberg-Marquardt algorithm, that ue the econd-order derivative of the phae of the STFT for the time-frequency localization of the ignal component to be weaker or tronger than with the claical reaignment. But although an efficient Time Frequency Repreentation TFR, the reaigned pectrogram i not directly invertible. To thi aim, the ynchroqueezing wa propoed, initially introduced for the Continuou Wavelet Tranform CWT [4] and later etended to the STFT [5, 6, 7, 8]. But all thee approache require the computation of everal STFT uing non-caual window which prevent their ue in real-time application. Thi reearch wa upported by the French ANR ASTRES project ANR-13-BS Thi problem can efficiently be olved by conidering a pecific cae of the STFT that can be recurively computed thank to the ue of a caual window function a in [9]. In the preent work, the firt- and econd-order derivative of the phae of the STFT are derived for thi particular cae and lead to a practical real-time implementation of both the claical and the Levenberg-Marquardt reaigned pectrogram, and to the firt to our knowledge real-time implementation of a ynchroqueezed STFT. Thi paper i organized a follow: in ection 2, the STFT i related to convolution product, to which reaignment and ynchroqueezing are applied. In ection 3, a particular analyi window i conidered to allow it implementation by caual recurive filter. Eperimental reult obtained with thee approache are preented in ection 4 and finally dicued with poible etenion in ection FILTER-BASED STFT, ITS REASSIGNMENT AND SYNCHROSQUEEZING The STFT F h t, ω M h t, ω e jφh t,ω, a defined for eample in [3], can be related to the linear convolution product between the analyzed ignal and the comple valued impule repone of a bandpa filter centered on ω, gt, ω ht e jωt, ht being a real-valued analyi window: y g t, ω + gτ,ωt τ dτ y g t, ω e jψg t,ω 1 F h t, ω e jωt M h t, ω e jφh t,ω+ωt 2 Thi implie that it magnitude M and it phae Φ can be deduced from the phae and magnitude of y g t, ω by M h t, ω y g t, ω and Φ h t, ωψ g t, ω ωt Rewording the claical reaignment The reaignment method, introduced in [1] and generalized by Auger and Flandrin in [2], aim at harpening a TFR. Thi improved localization of the ignal component i obtained by reaigning the value of an energy ditribution to coordinate that are cloer to the real upport of the analyzed ignal /16/$ IEEE 4880 ICASSP 2016

2 According to [1, 2], the reaignment operator of the pectrogram can be related to the phae of the STFT and thu can be reformulated uing the phae of y g t, ω, denoted Ψ g,a ˆtt, ω Φh t, ω t Ψg t, ω, 3 ˆωt, ω ω + Φh t, ω Ψg t, ω. 4 In practice, the partial derivative of Ψ g can be deduced from convolution product of the ignal with particular impule repone, a will be detailed in ection 2.3. Then, the reaigned pectrogram can imply be defined uing the reaigned coordinate a R h t, ω yt g,ω 2 δt ˆtt,ω δω ˆωt,ω dt dω, 5 R 2 where δt denote the Dirac ditribution Rewording the Levenberg-Marquardt reaignment Reflection on reaignment wa continued, and by analogy with the Levenberg-Marquardt root finding algorithm, new reaignment operator were derived [3] which allow to adjut the energy localization in the time-frequency plane through a damping parameter μ, which could be locally matched to the ignal content by or by a noie only/ignal+noie binary detector [10, 11]. Thee new reaignment operator can alo be epreed a a function of the phae Ψ g of y: g tt, ω t t R h 1 ωt, ω ω t, ω+μi 2 R h t, ω 6 Rt, h t ˆtt, ω ω ω ˆωt, ω t Rt, h R ω h R t, ω 2 Ψ g t, ω Ψ g ω Ψg h t, ω t, ω t, ω 2 Ψ g t, ω 2 2 t, ω 1 2 Ψ g t, ω 2 Ψ g where I 2 i the 2 2 identity matri. A a reult, the Levenberg-Marquardt reaigned pectrogram can be obtained by replacing ˆt, ˆω by t, ω in Eq Rewording the partial derivative of the phae A detailed in [3, 12], the firt- and econd-order derivative of the phae can be computed from STFT uing pecific window. If, unlike [3, 2], the phae Ψ and the impule repone g of the bandpa filter are ued intead of Φ and h, thee partial derivative can be computed a Ψ g y Dg t, ω t, ω Im yt, g 9 ω Ψ g t, ω Re y T g t, ω y g t, ω Ψ g y DT g t, ω t, ω Re yt, g ω 2 Ψ g 2 t, ω Im 2 Ψ g t, ω Im 2 y D2 g t, ω yt, g ω ydg y T 2 g t, ω yt, g ω t, ωy T g t, ω yt, g ω t, ω yt, g 12 ω y T g 2 t, ω yt, g 13 ω y Dg where Rez and Imz are repectively the real and imaginary part of a comple number z and y, g y T g, y Dg, y DT g, y T 2 g and y D2 g are the output of the filter uing repectively the impule repone gt, ω, T g tgt, ω, Dgt, ω g t, ω, DT gt, ω tgt, ω, T 2 gt, ωt 2 gt, ω and D 2 gt, ω 2 g t, ω Rewording the ynchroqueezed STFT A defined in Eq. 1, y g admit the following ignal recontruction formula t t 0 1 ht 0 + y g t, ω e jωt0 dω 2π, 14 when ω yt, g ω i integrable and when ht 0 0. Thee aumption are uppoed to be alway verified in the remainder of thi paper. Thi how that t can be recovered from y g with a time-delay t 0 0. Uually, a ymmetric non-caual window h i ued and the location of it the maimum i t 0 0. In ome cae, for eample when h0 0, the bet choice of t 0 depend on the window hape and hould be choen cloe to it maimum, a hown in ection 4. The ynchroqueezing method [4] i an alternative to the reaignment method that provide a harpened linear timefrequency tranform while allowing the ignal recontruction. The ynchroqueezed STFT [6] only ue the frequency coordinate reaignment operator and can be deduced from the ynthei formula 14 a Sy g t, ω yt, g ω e jω t 0 δ ω ˆωt, ω dω. 15 R Hence, Sy g t, ω 2 provide a harpen TFR and t can be etimated with a time-delay t 0 by ˆt t 0 1 Sy g dω t, ω ht 0 R 2π. 16 In practice, each ignal component can be recovered individually a propoed in [6] by retricting the integration area to the vicinity of each ridge. Eq. 15 claically ue the frequency reaignment operator defined by Eq. 4 and 9. We propoe to define a Levenberg-Marquardt Synchroqueezed STFT by replacing ˆω by ω in Eq. 15. Thi new ynchroqueezing proce allow to adjut the time-frequency localization of the ignal component while allowing ignal recontruction and mode etraction. 4881

3 3. TOWARDS A RECURSIVE IMPLEMENTATION A propoed in [9], y g can be recurively implemented if we ue for ht a caual recurive infinite impule repone filter t k 1 h k t T k k 1! e t/t Ut, 17 g k t, ω h k t e jωt t k 1 T k k 1! ept Ut 18 with p 1 T + jω, k 1 being the filter order, T the time pread of the window and Ut the Heaviide tep function Dicretization Uing the impule invariance method [13], the filter defined by Eq. 1 and 18 can be implemented a [9] G k z,ω T Z{g k t, ω} k 1 b i z i 1+ i0, 19 k a i z i i1 1 with b i L k k 1! B k 1,k i 1α i, α e pt, L T/T, Z{ft} + n0 fnt z n, a i A k,i α i, T being the ampling period. B k,i i j0 1j A k+1,j i +1 j k k k! denote the Eulerian number and A k,i i i!k i! the binomial coefficient. Hence, y k [n, m] y g k nt, 2πm MT can be computed from the ampled analyzed ignal [n] by a tandard recurive equation k 1 k y k [n, m] b i [n i] a i y k [n i, m] 20 i0 i1 where n Z and m 0, 1,..., M 1 are repectively the dicrete time and frequency indice. The z tranform of the other pecific impule repone can be epreed a function of G k z,ω at different order T Z{T g k t, ω} ktg k+1 z,ω 21 T Z{Dg k t, ω} 1 T G k 1z,ω+pG k z,ω 22 T Z{DTg k t, ω} k G k z,ω+pt G k+1 z,ω 23 T Z { T 2 g k t, ω } kk +1T 2 G k+2 z,ω 24 T Z { D 2 g k t, ω } 1 T 2 G k 2z,ω+ 2p T G k 1z,ω + p 2 G k z,ω. 25 Thee reult hold for any k 1 provided that G 0 z,ω G 1 z,ω 0. Eq. 21 and 22 generalize to any value of k ome reult already preented in [9], while Eq. 23 to 25 provide the dicrete-time linear ytem required by the Levenberg-Marquardt reaignment operator Recurive reaignment and ynchroqueezing The dicretization of the reaignment operator given by Eq. 3, 4, 6, 7 and 8 combined with epreion of the pecific window a function of G k z,ω lead to the following epreion of the dicrete-time reaigned coordinate [9] Re T 1 y T g [n, m] y[n, g m] T y Dg [n, m] y[n, g m] ˆn[n, m] n Round 26 M ˆm[n, m] Round 2π Im 27 and for the Levenberg-Marquardt reaignment we have 1 ñ[n, m] n Round T 1 Ψ g 1+μ 2 Ψ g Λ 1 T 2 2 Ψ g 2πm Λ 2 M T Ψ g 28 M m[n, m] m Round T 1 Ψ g T 2 2 Ψ g 2πΛ μ + 2 Ψ g m MT Ψ g 29 Λ 2π with Λ μ + 2 Ψ g μ +1 2 Ψ g + T 2 2 Ψ g T Ψ g 2 Thu, the reulting dicrete-time recurive reaigned pectrogram can be epreed a R k [n, m] n Z m 0 y k [n,m ] 2 δ [n ˆn[n,m ]] δ [m ˆm[n,m ]] 30 where δ[n] i the Kronecker delta and ˆn, ˆm can be replaced by ñ, m to obtain the Levenberg-Marquardt recurive reaigned pectrogram. The recurive ynchroqueezed STFT can be obtained a Sy k [n,m] m 0 y k [n,m ] e 2jπm n 0 M δ [m ˆm[n,m ]] 31 where n 0 t 0 /T can be choen a the time intant when the maimum of h k i reached i.e. n 0 k 1L. Thu, can be recovered from Sy k [n, m] by 1 ˆ[n n 0 ] Sy MT h k n 0 T k [n, m]. 32 m0 A the propoed reaignment and ynchroqueezing method are implemented not by FFT but by filter bank, the computational cot can be reduced by reducing the range of the computed frequency m to the epected frequency upport of the analyzed ignal. When uing the ynchroqueezed STFT, thi range reduction can alo be ued for mode retrieval or denoiing, a propoed in [6]. All reult preented in thi ection provide time-frequency analyi tool that are independent of the ignal ampling rate. 4882

4 4. EXPERIMENTAL RESULTS Recurive pectrogram k5, L7.00, SNR45 db Fig.1 compare the propoed recurive TFR obtained for a 500 ample long multicomponent real ignal made of two impule, one inuoid, one chirp and one inuoidally modulated inuoid. The animation included in the ubfigure, available with Adobe Reader c [14], how the reult obtained for M300 and a Signal-to-Noie Ratio SNR varying from 45 db down to 5 db obtained by addition of a white Gauian noie, with different value for k, L and μ. Thee animation clearly illutrate the improvement of the ignal component localization brought by the propoed TFR. For the Levenberg-Marquardt approach, the damping parameter μ hall not be choen too mall. Since only it frequency localization i improved, the ynchroqueezed STFT doe not improve the time localization of the impule. However it allow ignal recontruction and mode eparation. Table 1 how the enitivity to n 0, M and μ of the ignal Recontruction Quality Factor RQF, defined a a b c RQF 10log 10 n n [n] 2 [n] ˆ[n] 2 33 n RQF db M RQF db μ RQF db Table 1. Signal RQF of the recurive ynchroqueezed STFT computed for k 5, L 7 at SNR 45 db. Line a, computed for M 300, how that the bet RQF i obtained for n 0 k 1L, but decreaing n 0 only lightly decreae the RQF, while decreaing the ignal recontruction delay. Line b, computed for n 028, how that the RQF increae with M, but chooing M 200 only decreae the RQF from 6 db compared to M Line c, computed for n 028 and M 300, how that uing a Levenberg- Marquardt ynchroqueezed STFT with value of μ larger than 0.7 provide a better RQF than the claical ynchroqueezed STFT going to the RQF of the STFT, ee Eq. 14, equal to db, while maller value of μ provide a better ignal localization. 5. CONCLUSION We propoed a recurive implementation of the Levenberg- Marquardt reaignment and of the ynchroqueezing. Both are baed on the ue of caual recurive filter. Thank to the Levenberg-Marquardt algorithm, thee method can allow a uer to adjut the trength of the ignal localization in the time-frequency plane while allowing a ignal recontruction. Future work will conit in propoing related real-world data analyi application and to etend thi approach to econdorder ynchroqueezing [15]. A MATLAB implementation of the propoed method can be found on-line at [16]. 5 time ample a pectrogram obtained for everal k and SNR value. 5 5 Recur. rea. pectrogram k5, L7, SNR45 db time ample b claical and LM reaigned pectrogram. Recur. Sy k 2 k5, L7, SNR45 db time ample c quared modulu of claical- and LM-ynchroqueezed STFT. Fig. 1. Recurive time-frequency energy ditribution of a multicomponent ignal. All figure ue a linear gray cale and how TFR[n, m] α with α for the pectrogram and reaigned pectrogram, and α5 for the ynchroqueezing. 4883

5 6. REFERENCES [1] K. Kodera, C. de Villedary, and R. Gendrin, A new method for the numerical analyi of non-tationary ignal, Phyic of the Earth and Planetary Interior, vol. 12, pp , [2] F. Auger and P. Flandrin, Improving the readibility of time-frequency and time-cale repreentation by the reaignment method, IEEE Tran. Signal Proce., vol. 43, no. 5, pp , May [3] F. Auger, E. Chaande-Mottin, and P. Flandrin, Making reaignment adjutable: The Levenberg-Marquardt approach, in Proc. IEEE ICASSP 12, March 2012, pp [4] I. Daubechie and S. Mae, A nonlinear queezing of the continuou wavelet tranform baed on auditory nerve model, Wavelet in Medecine and Biology, pp , [5] F. Auger, P. Flandrin, Y-T Lin, S. McLaughlin, Meignen S., T. Oberlin, and H-T. Wu, Time-frequency reaignment and ynchroqueezing: an overview, IEEE Signal Proce. Mag., vol. 30, no. 6, pp , November [12] F. Auger, E. Chaande-Mottin, and P. Flandrin, On phae-magnitude relationhip in the hort-time Fourier tranform, IEEE Signal Proce. Lett., vol. 19, no. 5, pp , May [13] L.B. Jackon, A correction to impule invariance, IEEE Signal Proce. Lett., vol. 7, no. 10, pp , October [14] Adobe Reader, reader/, Acceed: January 14, [15] T. Oberlin, S. Meignen, and V. Perrier, Second-order ynchroqueezing tranform or invertible reaignment? toward ideal time-frequency repreentation, IEEE Tran. Signal Proce., vol. 63, no. 5, pp , March [16] ASTRES project MATLAB reource, http: // ANR_ASTRES/reource.html, Acceed: January 14, [6] S. Meignen, T. Oberlin, and S. McLaughlin, A new algorithm for multicomponent ignal analyi baed on ynchroqueezing: With an application to ignal ampling and denoiing, IEEE Signal Proce. Lett., vol. 60, no. 11, pp , Nov [7] Y. Guo, X. Fang, and X. Chen, A new improved ynchroqueezing tranform baed on adaptive hort time Fourier tranform, in Proc. IEEE Far Eat Forum on Nondetructive Evaluation/Teting FENDT, June 2014, pp [8] J. Thakur and H.-T. Wu, Synchroqueezing baed recovery of intantaneou frequency from nonuniform ample, SIAM J. Math. Anal., vol. 43, no. 5, pp , [9] G.K. Nilen, Recurive time-frequency reaignment, IEEE Tran. Signal Proce., vol. 57, no. 8, pp , Aug [10] Q.-H. Jo, J.-H. Chang, J.W. Shin, and N.S. Kim, Statitical model-baed voice activity detection uing upport vector machine, IET Signal Proceing, vol. 3, no. 3, pp , [11] J. Huillery, F. Millioz, and N. Martin, On the decription of pectrogram probabilitie with a chi-quared law, IEEE Tran. Signal Proce., vol. 56, no. 6, pp , June